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    Convolution PropertiesDSP for Scientists

    Department of PhysicsUniversity of Houston

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    Properties of Delta Function

    [n]: Identity for Convolution

    x[n] [n] =x[n]

    x[n] k[n] = kx[n]

    x[n] [n + s] =x[n + s]

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    Mathematical Properties of

    Convolution (Linear System)

    Commutative: a[n] b[n] = b[n] a[n]

    a[n] b[n] y[n]

    Then b[n] a[n] y[n]

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    Properties of Convolution

    Associative:{a[n] b[n]} c[n] = a[n] {b[n] c[n]}

    If

    a[n] b[n] c[n] y[n]

    Then a[n] b[n] c[n] y[n]

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    Properties of Convolution

    Distributive

    a[n]b[n] + a[n]c[n] = a[n]{b[n] + c[n]}

    If b[n]

    a[n] + y[n]

    c[n]

    Then

    a[n] b[n]+c[n] y[n]

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    Properties of Convolution

    Transference: between Input & Output

    Suppose x[n] * h[n] =y[n] IfL is a linear system,

    x1[n] =L{x[n]}, y1[n] =L{y[n]} Then

    x1[n] h[n]= y1[n]

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    Continue

    Ifx[n] h[n] y[n]

    Linear Same Linear

    System System

    Then

    x1[n] h[n] y1[n]

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    Special Convolution Cases

    Auto-Regression (AR) Model

    y[n] = k = 0, M - 1.h[k]x[n - k]

    For Example: y[n] =x[n] -x[n - 1] (first difference)

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    Special Convolution Cases

    Moving Average (MA) Model

    y[n] = b[0]x[n] + k= 1,M- 1b[k]y[n - k]

    For Example: y[n] =x[n] +y[n - 1] (Running Sum)

    AR and MA are Inverse to Each Other

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    Example

    For One-order Difference Equation (MAModel)

    y[n] = ay[n - 1] +x[n]

    Find the Impulse Response, if the system is

    (a) Causal

    (b) Anti-causal

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    Causal System Solution

    Input: [n] Output: h[n] For Causal system, h[n] = 0, n < 0

    h[0] = ah[-1] +[0] = 1

    h[1] = ah[0] + [1] = a

    h[n] = anu[n]

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    Anti-causal

    Input:x[n] = [n] Output:y[n] = h[n] For Anti-Causal system, h[n] = 0, n > 0

    y[n - 1] = (y[n] -x[n]) /a h[0] = (h[1] - [1]) /a = 0

    h[-1] = (h[0] -[0])/ a = - a

    -1

    h[- n] = - a-n h[n] = -anu[-n - 1]

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    Central Limit Theorem

    If a pulse-like signal is convoluted with

    itself many times, a Gaussian will be

    produced.

    a[n] 0

    a[n] a[n] a[n] a[n] = ???

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    Central Limit Theorem

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    Correlation !!!

    Cross-Correlation a[n] b[-n] = c[n]

    Auto-Correlation:

    a[n] a[-n] = c[n] Optimal Signal Detector (Not Restoration)

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    Correlation Detector

    - 1 5 - 1 0 - 5 0 5 1 0 1 5

    - 1

    - 0 . 8

    - 0 . 6

    - 0 . 4

    - 0 . 2

    0

    0 . 2

    0 . 4

    0 . 6

    0 . 8

    1

    - 2 5 - 2 0 - 1 5 - 1 0 - 5 0 5 1 0 1 5 2 0 2 5- 2

    - 1 . 5

    - 1

    - 0 . 5

    0

    0 . 5

    1

    1 . 5

    2

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    Correlation Results

    0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0

    -6

    -4

    -2

    0

    2

    4

    6

    8

    -5 -4 -3 -2 -1 0 1 2 3 4 5-0.6

    -0.4

    -0.2

    0

    0.2

    0.4

    0.6

    0.8

    1

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    Low-Pass Filter

    Filter h[n]:

    Cutoff the high-frequency components(undulation, pitches), smooth the signal

    h[n] 0, (- 1)n h[n] = 0, n = -,

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    Example: Lowpass

    0 50 100 150 200 250 300 350-60

    -40

    -20

    0

    20

    40

    60

    80

    10 0

    12 0

    14 0

    0 50 100 150 200 250 300 350

    -60

    -40

    -20

    0

    20

    40

    60

    80

    10 0

    12 0

    14 0

    -10 -8 -6 -4 -2 0 2 4 6 8 10-0.1

    0

    0.1

    0.2

    0.3

    0.4

    0.5

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    High-Pass Filter

    Filter g[n]:

    Remove the Average Value of Signal

    (Direct Current Components), Only

    Preserve the Quick Undulation Terms

    ng[n] = 0

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    Example: Highpass

    0 50 100 150 200 250 300 350-8

    -6

    -4

    -2

    0

    2

    4

    6

    8

    0 50 100 150 200 250 300 350

    -60

    -40

    -20

    0

    20

    40

    60

    80

    10 0

    12 0

    14 0

    -4 -3 -2 -1 0 1 2 3 4

    -0.6

    -0.4

    -0.2

    0

    0.2

    0.4

    0.6

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    Delta Function

    x[n] [n] =x[n]

    Do not Change Original Signal Delta function: All-Pass filter

    Further Change: Definition (Low-pass,

    High-pass, All-pass, Band-pass )

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